Some remarks on traces on the infinite-dimensional Iwahori--Hecke algebra

The infinite-dimensional Iwahori–Hecke algebras ${\mathcal{H}}_{\mathrm{\infty }}(q)$ are direct limits of the usual finite-dimensional Iwahori–Hecke algebras. They arise in a natural way as convolution algebras of bi-invariant functions on groups $\mathrm{G}\mathrm{L}\mathrm{B}({\mathbb{F}}_q)$ of infinite-dimensional matrices over finite-fields having only finite number of non-zero matrix elements under the diagonal. In 1988 Vershik and Kerov classified all indecomposable positive traces on ${\mathcal{H}}_{\mathrm{\infty }}(q)$. Any such trace generates a representation of the double ${\mathcal{H}}_{\mathrm{\infty }}(q)\otimes {\mathcal{H}}_{\mathrm{\infty }}(q)$ and of the double $\mathrm{G}\mathrm{L}\mathrm{B}({\mathbb{F}}_q)\times \mathrm{G}\mathrm{L}\mathrm{B}({\mathbb{F}}_q)$. We present constructions of such representations; the traces are some distinguished matrix elements. We also obtain some (simple) general statements on relations between unitary representations of groups and representations of convolution algebras of measures bi-invariant with respect to compact subgroups.